> On 13 Apr., 13:30, Dan <dan.ms.ch...@gmail.com> wrote:> > > The set of finite things has larger cardinality m than any finite> >> > > number n:> > > For all n exists m : m > n> >> > True, if m is indeed, a cardinality, and not a finite number .> >> > > But there is no cardinality m that is larger than every finite number> > > n> > > Exists m for all n : m > n> > > is wrong.> >> > http://en.wikipedia.org/wiki/Axiom_of_infinity> > The axiom of infinity, one of the basic axioms of set theory, says> > precisely that m exists .> > No.

Yes!> > > If you reject it,not only are we left unable to talk about real> > numbers ,banning them as invalid, we are left unable to talk about> > your binary tree .> > No. I assume it, when discussing here.

Then you assume an actual infinity, and cannot then impose its non-existence on what follows.

> Otherwise any discussion would> be in vain.

What makes you think the yours, at least, are not both vain and in vain?> >.> >> > In fact "all FISONs" are als an actually infinite set - and as such do> not exist.

Once you allow an actual infinitey, you can no longer deny it.

What exists is a FISON and for that there is a longer one.

And so on ad infinitum.> But already here we face a problem: The set> > 1> 2,1> 3,2,1> ...> > does not contain its limit |N as a line

Again WM does not comprehend that a strictly increasing sequence, like the above, is NEVER able to contain its limit.

That is the natural order of things which should hold even in the Stygian darknesses of Wolkenmuekenheim.

, but obviously contains its> limit |N in the first (and every other) column. On the other hand we> can prove, by the construction of this set, that everything that is in> the list, is in one line.

Thus again WM claims that a strictly increasing sequence, can contain (have a term equal to ) its limit.